We address the problem of filtering out localized time-frequency components in signals. The problem is formulated as a minimization of a suitable quadratic form, that involves a data fidelity term on the short-time Fourier transform outside the support of the undesired component, and an energy penalization term inside the support. The minimization yields a linear system whose solution can be expressed in closed form using Gabor multipliers. We provide an analysis of the solution and its approximation by truncated eigenvalue expansion, and illustrate its performance on synthetic mixtures of audio signals. The proposed approach outperforms approaches that are routinely used in applications.

We develop a time-scale synthesis-based probabilistic approach for the modeling of locally stationary signals. Inspired by our previous work, the model involves zero-mean, complex Gaussian wavelet coefficients, whose distribution varies as a function of time by time dependent translations on the scale axis. In a maximum a posteriori approach, we propose an estimator for the model parameters, namely the time-varying scale translation and an underlying power spectrum. The proposed approach is illustrated on a denoising example. It is also shown that the model can handle locally stationary signals with fast frequency variations, and provide in this case very sharp time-scale representations more concentrated than synchrosqueezed or reassigned wavelet transform.

A new approach for the analysis of nonstationary signals is proposed, with a focus on audio applications. Following earlier contributions, nonstationarity is modeled via stationarity-breaking operators acting on Gaussian stationary random signals. The focus is on time warping and amplitude modulation, and an approximate maximum-likelihood approach based on suitable approximations in the wavelet transform domain is developed. This paper provides theoretical analysis of the approximations, and introduces joint estimation of frequency, amplitude, and spectrum, a corresponding estimation algorithm. The latter is tested and validated on synthetic as well as real audio signal.

This paper proposes a new method for constructing and selecting of discriminant space-time-scale features for electroencephalogram (EEG) signal classification, suitable for Error Related Potentials (ErrP)detection in brain-computer interface (BCI). The method rests on a new variant of matrix-variate Linear Discriminant Analysis (LDA), and differs from previously proposed approaches in mainly three ways. First, a discrete wavelet expansion is introduced for mapping time-courses to time-scale coefficients, yielding time-scale localized features. Second, the matrix-variate LDA is modified in such a way that it yields an interesting duality property, that makes interpretation easier. Third, a space penalization is introduced using a surface Laplacian, so as to enforce spatial smoothness. The proposed approaches, termed D-MLDA and D-MPDA are tested on EEG signals, with the goal of detecting ErrP. Numerical results show that D-MPDA outperforms D-MLDA and other matrix-variate LDA techniques. In addition this method produces relevant features for interpretation in ErrP signals.

We propose here a new approach together with a corresponding class of algorithms for offline estimation of linear operators mapping input to output signals. The operators are modeled as multipliers, i.e., linear and diagonal operator in a frame or Bessel representation of signals (like Gabor, wavelets ...) and characterized by a transfer function. The estimation problem is formulated as a regularized inverse problem, and solved using iterative algorithms, based on gradient descent schemes. Various estimation problems, which differ by a choice for the regularization function, are studied in the case of Gabor multipliers. The transfer function actually provides a meaningful interpretation of the differences between the two signals or signal classes under consideration, and examples are discussed. Furthermore, examples of signal transformations with such Gabor transfer functions are also given.

Generalized versions of the entropic (Hirschman- Beckner) and support (Elad-Bruckstein) uncertainty principle are presented for frames representations. Moreover, a sharpened version of the support inequality is obtained by introducing a generalization of the coherence. In the finite-dimensional case and under certain conditions, minimizers of these inequalities are given. In addition,lpnorms inequalities are introduced as byproducts of the entropic inequalities.

We describe in this paper an audio denoising technique based on sparse linear regression with structured priors. The noisy signal is decomposed as a linear combination of atoms belonging to two modified discrete cosine transform (MDCT) bases, plus a residual part containing the noise. One MDCT basis has a long time resolution, and thus high frequency resolution, and is aimed at modeling tonal parts of the signal, while the other MDCT basis has short time resolution and is aimed at modeling transient parts (such as attacks of notes). The problem is formulated within a Bayesian setting. Conditional upon an indicator variable which is either 0 or 1, one expansion coefficient is set to zero or given a hierarchical prior. Structured priors are employed for the indicator variables; using two types of Markov chains, persistency along the time axis is favored for expansion coefficients of the tonal layer, while persistency along the frequency axis is favored for the expansion coefficients of the transient layer. Inference about the denoised signal and model parameters is performed using a Gibbs sampler, a standard Markov chain Monte Carlo (MCMC) sampling technique. We present results for denoising of a short glockenspiel excerpt and a long polyphonic music excerpt. Our approach is compared with unstructured sparse regression and with structured sparse regression in a single resolution MDCT basis (no transient layer). The results show that better denoising is obtained, both from signal-to-noise ratio measurements and from subjective criteria, when both a transient and tonal layer are used, in conjunction with our proposed structured prior framework.

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We study the behavior of hybrid random waveform models for audio signals, involving sparse random series of waveforms, with random coefficients. Similar approaches have been considered in the recent years. However, these do generally not rely on explicit models, and are of more "algorithmical" nature. The models we propose allow us to analyze mathematical properties of such signals and corresponding estimators, and derive estimation algorithms, which do not rely on complex optimization techniques

We describe a new method for estimating the degree of "transientness" and "tonality" of a class of compound signals involving simultaneously transient and harmonic features. The key assumption is that both transient and tonal layers admit sparse expansions, respectively in wavelet and local cosine bases. The estimation is performed using particular form of entropy (or theoretical dimension) functions. We provide theoretical estimates on the behavior of the proposed estimators, as well as numerical simulations. Audio signal coding provides a natural field of application.

A new approach for transients detection and estimation in the context of hybrid audio coding is presented. The basic idea is to use an orthogonal dyadic wavelet expansion, followed by hidden Markov tree modeling of wavelet coefficients. Coefficients may be cast as "transient type" or "residual type", and the estimated transient is reconstructed from the transient type coefficients only. The estimation procedure involves the classical two steps of hidden Markov models: parameters estimation and state estimation. The implementation of those two steps in the case of wavelet coefficient trees is discussed in some detail, and numerical results are given. The application to audio signal encoding is also discussed.

We present a novel way of implementing an audio coder aimed at good sound quality for very low bit-rates. The signal is first decomposed into a tonal part, transients and a stochastic part. The corresponding coefficients are quantized according to a psychoacoustic model, encoded with arithmetic coding and bit-packed. Some perceptually important high-level psychoacoustic attributes are also encoded, and used by the decoder in a post-processing unit.

The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated. In particular, it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws. Applications to spectral line estimations and matched filtering are briefly discussed.>

Two methods for computing the (asymptotic) electromagnetic scattering by arbitrarily-shaped targets are described: the Eikonal model, valid for objects whose overall size is much greater than the incident wavelength (based on a quantum partial-wave expansion in the scalar approximation); and the low-energy model, valid for objects whose overall size is comparable to the incident wavelength (based on a discretization of the scattering object). Their numerical results are compared to those of exact theories in the cases of perfect spheres and spheroids. Comparisons to experimental data for irregular objects (Eikonal model) and dielectric helices (low-energy model) are described, as well as theoretical predictions for rough scatterers.>